The new dice pdf is out!

The pdf is filled with interesting solutions and if you are a lover of generating functions, recurrences, Markov chains and computation, all the things I babble about in Computer Math then you will love the way a real artist uses them.

Thanks Matt!

Now, about that solution to #10...

Rumpole once sarcastically said that he always found that any knowledge of the law is a severe disadvantage. Likewise, knowledge of mathematics can ruin you when you come across a problem like that. Two things hindered me in solving that problem, lack of courage, I could not bring myself to state it as a 730 state Markov chain and a stubborn reliance on using the notation and tools of linear algebra to deal with it!

In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Re: The new dice pdf is out!

My answer to the last question in problem #10 regarding the case in which we want a number to appear 3 times before stopping:

First, we need to find the probability P(X=i) that the first number that appears 3 times is in the i-th postition:

For the first number that appears 3 times to be in the i-th position, it must also appear exactly twice in the previous i-1 places, and all other number can appear at most 2 time. This means, also, that, by the Pigeon hole principle, i can take values between 3 and 13, only.

Now, the number of ways we can choose two numbers from the first i-1 is

, and the probability that they are the same as the i-th number is .

The rest of the numbers can appear at most 2 times, so we can use the GF

to model them. Because the total number of other numbers in the first i-1 is i-3, and because there are only 5 possible choices for those i-3 numbers, we will model this with the GF , whose x^(i-3) term's coefficient is equal to the number of ways of distributing the i-3 numbers among the five possibilities. But this only gives us the possible distribution of those i-3 numbers, without caring for theor order in the throw sequence. To account for that we need a factor of (i-3)!. So, finally, the number of ways the i-3 numbers can be distributed is . Also, the probability that those numbers will have the values assigned to them is

Combining this with the previous results gives us the probability of

.

To get the expected value, we use the standar formula E(X)=∑ i*P(X=i):

.

Now we can use a CAS to find the value of this sum, or to get a closed form.

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: The new dice pdf is out!

Thanks!

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: The new dice pdf is out!

That is great! I will wait for him to answer my email now.

In the meantime, we can work on a better form for that, or an asymptotic form.

Last edited by anonimnystefy (2013-02-16 10:25:44)

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: The new dice pdf is out!

It seems that the inner sum has a closed form in terms of the Gamma and the upper incomplete Gamma function.

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: The new dice pdf is out!

That sum is troubling me. Is there anyway to get rid of it?

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: The new dice pdf is out!

The inner sum.

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: The new dice pdf is out!

I don't understand you.

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: The new dice pdf is out!

By the way, Matt answere asking what my method was. I answered him and told him I could send him the full proof and that I have found a generalization.

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: The new dice pdf is out!

I will if he says he's interested.in seeing it.

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: The new dice pdf is out!

Yes. He asked for the method, but not the whole solution.

By the way, you really should not lie.

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: The new dice pdf is out!

Lie about this:

bobbym wrote:

Hi gAr;

I am sorry, but on this one Mr anonimnystefy has left me dumber than a burned out stump. Apparently his method has fried my brain. I am so confused I can not solve 3 + x = 9. Can you show me what you want done? You can use codecogs for the latexing.

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.

Re: The new dice pdf is out!

Maybe it's because of your back.

Have you got an idea for clearing up the formula up there.

Here lies the reader who will never open this book. He is forever dead.Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and PunishmentThe knowledge of some things as a function of age is a delta function.